It is easy to prove that the completion of your space can be any separable metric space 
where metric spheres are nowhere dense.
>Does not it scare you?

**Answer to the comment:**
Not all of your spaces can be embedded into a metric tree.

BTW, there is a nice characterization of subsets in a metric tree:
$$ | x - y | + | a - b | \le |x-a| + |x-b| + |y-a| + |y-b|$$
for all points $x,y,a,b$
(here $|x-y|$ denotes the distance from $x$ to $y$).
This inequality  can be also thought as a discrete analog of CAT[−∞] inequality.